Radon-Nikodym property in locally convex spaces.

Abstract

Our object in this thesis is to study the Radon-Nikodym property (RNP) in the class of locally convex spaces (l.c.s). We divide this work into four chapters. Chapter one is an introductory chapter, containing various definitions, theorems and notations which are needed in the other chapters, such as vector measure, RNP, bounded variation of vector measures, dentability, etc. Chapter 2 The Liapounoff convexity theorem and the Uhl generalization of this theorem on the class of Banach spaces which are either reflexive or separable dual spaces are given in chapter two. We give two examples due to Uhl to show that this generalization cannot be improved under the current hypotheses.Our goal in chapter two is to give a generalization of the Uhl convexity theorem on the range of vector measures in the class of locally convex spaces with a Radon-Nikodym derivative. Chapter 3 Rieffel proved the fundamental Radon-Nikodym Theorem (RNT) for Banach spaces. Since then, various efforts have been made to extend Rieffels (RNT) to locally convex spaces.Saab extended Rieffels Theorem in the class of quasi-complete locally convex spaces with property that every bounded set is metrizable. Our goal is to give a generalization of the result of Saab using the same technique he used for general locally convex spaces and this is contained in chapter three. The equivalence between the Radon-Nikodym property and Bishop-Phelps property (BPP) in the class of Banach spaces was proved by J. Bourgin . We prove the relation between these properties in the class of locally convex spaces with the property that every bounded set is metrizable. To prove this we use a new definition of (BPP) in locally convex spaces and the result of Saab.